44.1.9 problem 1(j)

Internal problem ID [9067]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Section 1.2 THE NATURE OF SOLUTIONS. Page 9
Problem number : 1(j)
Date solved : Tuesday, September 30, 2025 at 06:03:28 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y^{\prime }&=\frac {x y}{x^{2}+y^{2}} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 16

ode:=diff(y(x),x) = x*y(x)/(x^2+y(x)^2); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {\frac {1}{\operatorname {LambertW}\left (c_1 \,x^{2}\right )}}\, x \]

✓ Mathematica. Time used: 5.48 (sec). Leaf size: 49

ode=D[y[x],x]==(x*y[x])/(x^2+y[x]^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {x}{\sqrt {W\left (e^{-2 c_1} x^2\right )}}\\ y(x)&\to \frac {x}{\sqrt {W\left (e^{-2 c_1} x^2\right )}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.669 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*y(x)/(x**2 + y(x)**2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = e^{C_{1} + \frac {W\left (x^{2} e^{- 2 C_{1}}\right )}{2}} \]